Let (A, A ′ , A H ) be the triple of hyperplane arrangements. We show that the freeness of A H and the division of χ(A; t) by χ(A H ; t) imply the freeness of A. This "division theorem" improves the famous addition-deletion theorem, and it has several applications, which include a definition of "divisionally free arrangements". It is a strictly larger class of free arrangements than the classical important class of inductively free arrangements. Also, whether an arrangement is divisionally free or not is determined by the combinatorics. Moreover, we show that a lot of recursively free arrangements, to which almost all known free arrangements are belonging, are divisionally free. Main resultsLet V be an ℓ-dimensional vector space over an arbitrary field K with ℓ ≥ 1, S = Sym(V * ) = K[x 1 , . . . , x ℓ ] its coordinate ring and Der S := ⊕ ℓ i=1 S∂ x i the module of K-linear S-derivations. A hyperplane arrangement A is a finite set of hyperplanes in V . We say that A is central if every hyperplane is linear. In this article every arrangement is central unless otherwise specified. In the central cases, we fix a linear form α H ∈ V * such that ker(α H ) = H for each H ∈ A. An ℓ-arrangement is an arrangement in an ℓ-dimensional vector space. Let L(A) := {∩ H∈B H | B ⊂ A} be an intersection lattice. L(A) has a partial order by reverse inclusion, which equips L(A) with a poset structure. For X ∈ L(A), define the localization A X of A at X by A X := {H ∈ A | H ⊃ X}, which is a subarrangement of A. Also, the restriction A X of A onto X is defined by A X := {H ∩ X | H ∈ A \ A X }, which is an *
A Weyl arrangement is the arrangement defined by the root system of a finite Weyl group. When a set of positive roots is an ideal in the root poset, we call the corresponding arrangement an ideal subarrangement. Our main theorem asserts that any ideal subarrangement is a free arrangement and that its exponents are given by the dual partition of the height distribution, which was conjectured by Sommers-Tymoczko. In particular, when an ideal subarrangement is equal to the entire Weyl arrangement, our main theorem yields the celebrated formula by Shapiro, Steinberg, Kostant, and Macdonald. The proof of the main theorem is classification-free. It heavily depends on the theory of free arrangements and thus greatly differs from the earlier proofs of the formula.
We study a relation between roots of characteristic polynomials and intersection points of line arrangements. Using these results, we obtain a lot of applications for line arrangements. Namely, we give (i) a generalized addition theorem for line arrangements, (ii) a generalization of Faenzi-Vallès' theorem over a field of arbitrary characteristic, (iii) a partial result on the conjecture of Terao for line arrangements, and (iv) a new sufficient condition for freeness over finite fields. Also, a higher dimensional version of our main results are considered.2000 Mathematics Subject Classification. Primary, 32S22.
The addition-deletion theorems for hyperplane arrangements, which were originally shown in [T1], provide useful ways to construct examples of free arrangements. In this article, we prove addition-deletion theorems for multiarrangements. A key to the generalization is the definition of a new multiplicity, called the Euler multiplicity, of a restricted multiarrangement. We compute the Euler multiplicities in many cases. Then we apply the addition-deletion theorems to various arrangements including supersolvable arrangements and the Coxeter arrangement of type A3 to construct free and non-free multiarrangements. IntroductionLet A be a hyperplane arrangement, or simply an arrangement. In other words, A is a finite collection of hyperplanes in an ℓ-dimensional vector space V over a field K. A multiarrangement, which was introduced by Ziegler in [Z], is a pair (A, m) consisting of a hyperplane arrangement A and a multiplicity m : A → Z >0 . Define |m| = H∈A m(H). A multiarrangement (A, m) such that m(H) = 1 for all H ∈ A is just a hyperplane arrangement, and is sometimes called a simple arrangement.Let {x 1 , . . . , x ℓ } be a basis forWhen each H ∈ A contains the origin, we say that A is central. Throughout this article, assume that every arrangement is central. Let Der K (S) denote the set of K-linear derivations from S to itself. For each H ∈ A we choose a defining formIf D(A, m) is a free S-module we say that (A, m) is a free multiarrangement. In his groundbreaking paper [Z], Ziegler writes ". . . the theory of multiarrangements and their freeness is not yet in a satisfactory state. In particular, we do not know any addition/deletion theorem . . . ." It is exactly the subject of this article. Namely, we generalize the addition-deletion theorems for simple arrangements [T1] to multiarrangements in this article. Let (A, m) be a nonempty multiarrangement and ℓ ≥ 2. Fix a hyperplane H 0 ∈ A and let α 0 be a defining form for H 0 . To state the addition-deletion theorems for multiarrangements we need to define the deletion (A ′ , m ′ ) and the restriction (A ′′ , m ′′ ). First, we define the deletion as follows:Next we define the restriction (A ′′ , m ′′ ). Letwhich is an arrangement on H 0 . We, however, have more than one choice to define a multiplicity m ′′ . The definition of a suitable multiplicity m ′′ is crucial. The canonical definition is probablywhich is purely combinatorial and was used in [Y1, Y2, Z] effectively. In this article, however, in order to serve our purposes, we introduce a new multiplicity m * , called the Euler multiplicity, whose definition is algebraic rather than combinatorial.For X ∈ A ′′ define A X = {H ∈ A | X ⊂ H} and m X = m | AX .
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.